Circuits, Systems, and Signal Processing

, Volume 38, Issue 7, pp 3321–3339 | Cite as

Adaptive Exponential State Estimation for Markovian Jumping Neural Networks with Multi-delays and Lévy Noises

  • Qiaoyu Chen
  • Dongbing TongEmail author
  • Wuneng ZhouEmail author
  • Yuhua XuEmail author
Short Paper


This paper discusses the adaptive exponential state estimation problem of neutral-type neural networks with multi-delays and Lévy noises. The M-matrix method being different from other methods, such as the LMIs method, has been applied to deal with the problem. According to the M-matrix method, some state estimation criteria for neural networks concerning neutral-type delays and no neutral-type delays are acquired to ensure the adaptive exponential estimation. Finally, a simulation example is offered to show the advantages of the theoretical results.


Neural networks Exponential state estimation Adaptive control Multi-delays Lévy noises 



This study was funded by the National Natural Science Foundation of China (61673257; 11501367; 61573095; 61673221); the Natural Science Foundation of Jiangsu Province (BK20181418); the fifteenth batch of Six Talent Peaks Project in Jiangsu Province (DZXX-019).


  1. 1.
    D. Applebaum, Lévy Processes and Stochastic Calculus, 2nd edn. (Cambridge University Press, Cambridge, 2009)CrossRefzbMATHGoogle Scholar
  2. 2.
    H. Chen, P. Shi, C.C. Lim, Exponential synchronization for Markovian stochastic coupled neural networks of neutral-type via adaptive feedback control. IEEE Trans. Neural Netw. Learn. Syst. 28(7), 1618–1632 (2017)MathSciNetCrossRefGoogle Scholar
  3. 3.
    P. Cheng, Y. Qi, K. Xin, J. Chen, L. Xie, Energy-efficient data forwarding for state estimation in multi-hop wireless sensor networks. IEEE Trans. Autom. Control 61(5), 1322–1327 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    A. Gómez-Expósito, C. Gómez-Quiles, I. Džafić, State estimation in two time scales for smart distribution systems. IEEE Trans. Smart Grid 6(1), 421–430 (2015)CrossRefGoogle Scholar
  5. 5.
    L.V. Hien, D.T. Son, H. Trinh, On global dissipativity of nonautonomous neural networks with multiple proportional delays. IEEE Trans. Neural Netw. Learn. Syst. 29(1), 225–231 (2018)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Y. Ji, F. Ding, Multiperiodicity and exponential attractivity of neural networks with mixed delays. Circuits Syst. Signal Process. 36(6), 2558–2573 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    G. Jumarie, Modeling fractional stochastic systems as non-random fractional dynamics driven by Brownian motions. Appl. Math. Model. 32(5), 836–859 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    K. Lee, Y. Kim, S. Chong, I. Rhee, Y. Yi, N.B. Shroff, On the critical delays of mobile networks under Lévy walks and Lévy flights. IEEE/ACM Trans. Netw. 21(5), 1621–1635 (2013)CrossRefGoogle Scholar
  9. 9.
    M. Liu, S. Zhang, Z. Fan, S. Zheng, W. Sheng, Exponential \(H_\infty \) synchronization and state estimation for chaotic systems via a unified model. IEEE Trans. Neural Netw. Learn. Syst. 24(7), 1114–1126 (2013)CrossRefGoogle Scholar
  10. 10.
    X. Liu, M. Dong, K. Ota, P. Hung, A. Liu, Service pricing decision in cyber-physical systems: insights from game theory. IEEE Trans. Serv. Comput. 9(2), 186–198 (2016)CrossRefGoogle Scholar
  11. 11.
    X. Liu, Z. Zeng, S. Wen, Implementation of memristive neural network with full-function pavlov associative memory. IEEE Trans. Circuits Syst. Regul. Pap. 63(9), 1454–1463 (2016)MathSciNetCrossRefGoogle Scholar
  12. 12.
    R. Lu, P. Shi, H. Su, Z.G. Wu, J. Lu, Synchronization of general chaotic neural networks with nonuniform sampling and packet missing: a switched system approach. IEEE Trans. Neural Netw. Learn. Syst. 29(3), 523–533 (2016)MathSciNetCrossRefGoogle Scholar
  13. 13.
    X. Mao, Stochastic Differential Equations and Applications (Elsevier, Amsterdam, 2007)zbMATHGoogle Scholar
  14. 14.
    J.L. Mathieu, S. Koch, D.S. Callaway, State estimation and control of electric loads to manage real-time energy imbalance. IEEE Trans. Power Syst. 28(1), 430–440 (2013)CrossRefGoogle Scholar
  15. 15.
    E. Nadal, J.V. Aguado, E. Abisset-Chavanne, F. Chinesta, R. Keunings, E. Cueto, A physically-based fractional diffusion model for semi-dilute suspensions of rods in a Newtonian fluid. Appl. Math. Model. 51, 58–67 (2017)MathSciNetCrossRefGoogle Scholar
  16. 16.
    P. Nowak, M. Pawłowski, Option pricing with application of Lévy processes and the minimal variance equivalent martingale measure under uncertainty. IEEE Trans. Fuzzy Syst. 25(2), 402–416 (2017)CrossRefGoogle Scholar
  17. 17.
    S. Peng, F. Li, L. Wu, C.C. Lim, Neural network-based passive filtering for delayed neutral-type semi-Markovian jump systems. IEEE Trans. Neural Netw. Learn. Syst. 28(9), 2101–2114 (2017)MathSciNetGoogle Scholar
  18. 18.
    Que, H., Fang, M., Wu, Z.G., Su, H., Huang, T., Zhang, D.: Exponential synchronization via aperiodic sampling of complex delayed networks. IEEE Trans. Syst. Man Cybern. Syst. (2018).
  19. 19.
    R. Sakthivel, P. Vadivel, K. Mathiyalagan, A. Arunkumar, M. Sivachitra, Design of state estimator for bidirectional associative memory neural networks with leakage delays. Inf. Sci. 296, 263–274 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Y. Shu, X.G. Liu, Y. Liu, J.H. Park, Improved results on guaranteed generalized \(H_2\) performance state estimation for delayed static neural networks. Circuits Syst. Signal Process. 36(8), 3114–3142 (2017)CrossRefzbMATHGoogle Scholar
  21. 21.
    Tao, J., Wu, Z.G., Su, H., Wu, Y., Zhang, D.: Asynchronous and resilient filtering for Markovian jump neural networks subject to extended dissipativity. IEEE Trans. Cybern. (2018).
  22. 22.
    D. Tong, Q. Chen, Delay and its time-derivative-dependent model reduction for neutral-type control system. Circuits Syst. Signal Process. 36(6), 2542–2557 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    D. Tong, P. Rao, Q. Chen, M.J. Ogorzalek, X. Li, Exponential synchronization and phase locking of a multilayer Kuramoto-oscillator system with a pacemaker. Neurocomputing 308, 129–137 (2018)CrossRefGoogle Scholar
  24. 24.
    D. Tong, W. Zhou, X. Zhou, J. Yang, L. Zhang, Y. Xu, Exponential synchronization for stochastic neural networks with multi-delayed and Markovian switching via adaptive feedback control. Commun. Nonlinear Sci. Numer. Simulat. 29(1–3), 359–371 (2015)MathSciNetCrossRefGoogle Scholar
  25. 25.
    J.L. Wang, H.N. Wu, T. Huang, S.Y. Ren, J. Wu, Pinning control for synchronization of coupled reaction-diffusion neural networks with directed topologies. IEEE Trans. Syst. Man Cybern. Syst. 46(8), 1109–1120 (2016)CrossRefGoogle Scholar
  26. 26.
    L. Wang, Z. Wang, T. Huang, G. Wei, An event-triggered approach to state estimation for a class of complex networks with mixed time delays and nonlinearities. IEEE Trans. Cybern. 46(11), 2497–2508 (2016)CrossRefGoogle Scholar
  27. 27.
    Z. Wang, K.J. Burnham, Robust filtering for a class of stochastic uncertain nonlinear time-delay systems via exponential state estimation. IEEE Trans. Signal Process. 49(4), 794–804 (2001)CrossRefGoogle Scholar
  28. 28.
    C. Xu, P. Li, Global exponential convergence of fuzzy cellular neural networks with leakage delays, distributed delays and proportional delays. Circuits Syst. Signal Process. 37(1), 163–177 (2018)MathSciNetCrossRefGoogle Scholar
  29. 29.
    J. Yang, W. Zhou, P. Shi, X. Yang, X. Zhou, H. Su, Synchronization of delayed neural networks with Lévy noise and Markovian switching via sampled data. Nonlinear Dyn. 81(3), 1179–1189 (2015)CrossRefzbMATHGoogle Scholar
  30. 30.
    J. Yang, W. Zhou, X. Yang, X. Hu, L. Xie, \(p\)th moment asymptotic stability of stochastic delayed hybrid systems with Lévy noise. Int. J. Control 88(9), 1726–1734 (2015)CrossRefzbMATHGoogle Scholar
  31. 31.
    X. Yang, J. Cao, J. Liang, Exponential synchronization of memristive neural networks with delays: interval matrix method. IEEE Trans. Neural Netw. Learn. Syst. 28(8), 1878–1888 (2017)MathSciNetCrossRefGoogle Scholar
  32. 32.
    G. Zhang, Y. Song, T.Q. Zhang, Stochastic resonance in a single-well system with exponential potential driven by Lévy noise. Chin. J. Phys. 55(1), 85–95 (2017)MathSciNetCrossRefGoogle Scholar
  33. 33.
    H. Zhang, Z. Wang, D. Liu, Global asymptotic stability of recurrent neural networks with multiple time-varying delays. IEEE Trans. Neural Netw. 19(5), 855–873 (2008)CrossRefGoogle Scholar
  34. 34.
    W. Zhang, Y. Tang, Y. Liu, J. Kurths, Event-triggering containment control for a class of multi-agent networks with fixed and switching topologies. IEEE Trans. Circuits Syst. Regul. Pap. 64(3), 619–629 (2017)CrossRefGoogle Scholar
  35. 35.
    W. Zhang, T. Yang, T. Huang, J. Kurths, Sampled-data consensus of linear multi-agent systems with packet losses. IEEE Trans. Neural Netw. Learn. Syst. 28(11), 2516–2527 (2016)MathSciNetCrossRefGoogle Scholar
  36. 36.
    J. Zhou, X. Ding, L. Zhou, W. Zhou, J. Yang, D. Tong, Almost sure adaptive asymptotically synchronization for neutral-type multi-slave neural networks with Markovian jumping parameters and stochastic perturbation. Neurocomputing 214, 44–52 (2016)CrossRefGoogle Scholar
  37. 37.
    L. Zhou, Q. Zhu, Z. Wang, W. Zhou, H. Su, Adaptive exponential synchronization of multislave time-delayed recurrent neural networks with Lévy noise and regime switching. IEEE Trans. Neural Netw. Learn. Syst. 28(12), 2885–2898 (2017)MathSciNetCrossRefGoogle Scholar
  38. 38.
    W. Zhou, Y. Gao, D. Tong, C. Ji, J. Fang, Adaptive exponential synchronization in \(p\)th moment of neutral-type neural networks with time delays and Markovian switching. Int. J. Control Autom. Syst. 11(4), 845–851 (2013)CrossRefGoogle Scholar
  39. 39.
    W. Zhou, D. Tong, Y. Gao, C. Ji, H. Su, Mode and delay-dependent adaptive exponential synchronization in \(p\)th moment for stochastic delayed neural networks with Markovian switching. IEEE Trans. Neural Netw. Learn. Syst. 23(4), 662–668 (2012)CrossRefGoogle Scholar
  40. 40.
    W. Zhou, Q. Zhu, P. Shi, H. Su, J. Fang, L. Zhou, Adaptive synchronization for neutral-type neural networks with stochastic perturbation and Markovian switching parameters. IEEE Trans. Cybern. 44(12), 2848–2860 (2014)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of Statistics and MathematicsShanghai Lixin University of Accounting and FinanceShanghaiChina
  2. 2.School of Electronic and Electrical EngineeringShanghai University of Engineering ScienceShanghaiChina
  3. 3.College of Information Sciences and TechnologyDonghua UniversityShanghaiChina
  4. 4.School of FinanceNanjing Audit UniversityJiangsuChina

Personalised recommendations